Bounds on exponential moments of hitting times for reflected processes on the positive orthant

Abstract

We first consider a multi-dimensional reflected fractional Brownian motion process on the positive orthant with the Hurst parameter H∈(0,1). In particular, when H>1/2, this model serves to approximate fluid stochastic network models fed by a big number of heavy tailed ON/OFF sources in heavy traffic. Assuming the initial state lies outside some compact set, we establish that the exponential moment of the first hitting time to the compact set has a lower bound with an exponential growth rate in terms of the magnitude of the initial state. We extend this result to the case for reflected processes driven by a class of stable Lévy motions, which arise as approximations to cumulative network traffic over a time period. For the case of H=1/2, under a natural stability condition on the reflection directions and drift vector, we obtain a matching upper bound on exponential moments of hitting times, which grows at an exponential rate in terms of the initial condition of the process. We also show that such an upper bound is valid for reflected processes driven by general light-tailed Lévy processes.